युग्म (kx+3y=7) और (4x+6y=9) का अद्वितीय हल कब होगा?
When will the pair (kx+3y=7) and (4x+6y=9) have a unique solution?
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#linear-equations
#solvability
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A (k=2)
B (k=0)
C \(k\neq2\)
D \(k\neq4\)
Explanation opens after your attempt
Correct Answer
C. \(k\neq2\)
Step 1
Concept
For a unique solution, coefficient ratios must not be equal. Here \(k\neq2\); check ratios directly in exams.
Step 2
Why this answer is correct
The correct answer is C. \(k\neq2\). For a unique solution, coefficient ratios must not be equal. Here \(k\neq2\); check ratios directly in exams.
Step 3
Exam Tip
अद्वितीय हल के लिए गुणांकों के अनुपात बराबर नहीं होने चाहिए। यहाँ \(k\neq2\) परीक्षा में सीधे अनुपात से जाँचें।
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युग्म (2x+ky=8) और (5x+10y=12) के लिए कोई हल न होने की शर्त क्या है?
What is the condition for no solution for the pair (2x+ky=8) and (5x+10y=12)?
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A (k=5)
B (k=4)
C (k=10)
D (k=2)
Explanation opens after your attempt
Step 1
Concept
For no solution, coefficient ratios are equal but the constant ratio is different. At (k=4), this condition holds.
Step 2
Why this answer is correct
The correct answer is B. (k=4). For no solution, coefficient ratios are equal but the constant ratio is different. At (k=4), this condition holds.
Step 3
Exam Tip
कोई हल नहीं के लिए गुणांकों के अनुपात समान और स्थिर पद का अनुपात अलग होना चाहिए। (k=4) पर यही स्थिति बनती है।
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युग्म ((m-1 )x+6y=10) और (3x+9y=15) के अनंत हलों के लिए (m) का मान क्या होगा?
For infinitely many solutions of ((m-1 )x+6y=10) and (3x+9y=15), what is the value of (m)?
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#solvability
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A (m=1)
B (m=2)
C (m=3)
D (m=5)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, all three ratios are equal. From \(\frac{m-1}{3}=\frac{6}{9}=\frac{10}{15}\), (m=3).
Step 2
Why this answer is correct
The correct answer is C. (m=3). For infinitely many solutions, all three ratios are equal. From \(\frac{m-1}{3}=\frac{6}{9}=\frac{10}{15}\), (m=3).
Step 3
Exam Tip
अनंत हलों के लिए तीनों अनुपात समान होते हैं। यहाँ \(\frac{m-1}{3}=\frac{6}{9}=\frac{10}{15}\) से (m=3) मिलता है।
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युग्म ((p+2)x+5y=11) और (6x+10y=13) का अद्वितीय हल कब होगा?
When will the pair ((p+2)x+5y=11) and (6x+10y=13) have a unique solution?
#class10
#linear-equations
#solvability
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A (p=1)
B (p=3)
C (p=0)
D \(p\neq1\)
Explanation opens after your attempt
Correct Answer
D. \(p\neq1\)
Step 1
Concept
For a unique solution, \(\frac{p+2}{6}\neq\frac{5}{10}\). Hence \(p\neq1\) is correct.
Step 2
Why this answer is correct
The correct answer is D. \(p\neq1\). For a unique solution, \(\frac{p+2}{6}\neq\frac{5}{10}\). Hence \(p\neq1\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{p+2}{6}\neq\frac{5}{10}\) होना चाहिए। इसलिए \(p\neq1\) सही शर्त है।
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युग्म (9x-3y=6) और ((q+3)x-qy=10) के लिए कोई हल न होने पर (q) क्या होगा?
If (9x-3y=6) and ((q+3)x-qy=10) have no solution, what is (q)?
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A \(q=\frac{3}{2}\)
B (q=2)
C (q=3)
D \(q=\frac{1}{2}\)
Explanation opens after your attempt
Correct Answer
A. \(q=\frac{3}{2}\)
Step 1
Concept
For distinct parallel lines, coefficient ratios are equal and constant ratio is different. Solving gives \(q=\frac{3}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(q=\frac{3}{2}\). For distinct parallel lines, coefficient ratios are equal and constant ratio is different. Solving gives \(q=\frac{3}{2}\).
Step 3
Exam Tip
समांतर अलग रेखाओं के लिए गुणांक अनुपात समान और स्थिर अनुपात अलग होता है। हल करने पर \(q=\frac{3}{2}\) मिलता है।
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युग्म (4x+(r-2)y=14) और (10x+15y=35) के अनंत हलों के लिए (r) का मान क्या है?
What is the value of (r) for infinitely many solutions of (4x+(r-2)y=14) and (10x+15y=35)?
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A (r=6)
B (r=8)
C (r=10)
D (r=12)
Explanation opens after your attempt
Step 1
Concept
In infinitely many solutions, \(\frac{4}{10}=\frac{r-2}{15}=\frac{14}{35}\). This gives (r=8).
Step 2
Why this answer is correct
The correct answer is B. (r=8). In infinitely many solutions, \(\frac{4}{10}=\frac{r-2}{15}=\frac{14}{35}\). This gives (r=8).
Step 3
Exam Tip
अनंत हलों में \(\frac{4}{10}=\frac{r-2}{15}=\frac{14}{35}\) होता है। इससे (r=8) प्राप्त होता है।
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युग्म (ax+7y=13) और (5x+by=9) के अद्वितीय हल की सही शर्त कौन-सी है?
Which condition gives a unique solution for (ax+7y=13) and (5x+by=9)?
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A (ab=35)
B (a=b)
C \(ab\neq35\)
D (a+b=35)
Explanation opens after your attempt
Correct Answer
C. \(ab\neq35\)
Step 1
Concept
For a unique solution, determinant (D=ab-35) must be non-zero. So \(ab\neq35\) is correct.
Step 2
Why this answer is correct
The correct answer is C. \(ab\neq35\). For a unique solution, determinant (D=ab-35) must be non-zero. So \(ab\neq35\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए सारणिक (D=ab-35) शून्य नहीं होना चाहिए। इसलिए \(ab\neq35\) सही है।
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युग्म (6x-8y=16) और (sx-12y=24) के अनंत हलों के लिए (s) का मान क्या होगा?
For infinitely many solutions of (6x-8y=16) and (sx-12y=24), what is (s)?
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A (s=6)
B (s=8)
C (s=9)
D (s=12)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, all ratios must be equal. From \(\frac{6}{s}=\frac{-8}{-12}=\frac{16}{24}\), (s=9).
Step 2
Why this answer is correct
The correct answer is C. (s=9). For infinitely many solutions, all ratios must be equal. From \(\frac{6}{s}=\frac{-8}{-12}=\frac{16}{24}\), (s=9).
Step 3
Exam Tip
अनंत हलों के लिए सभी अनुपात बराबर होने चाहिए। \(\frac{6}{s}=\frac{-8}{-12}=\frac{16}{24}\) से (s=9) है।
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युग्म (3x+4y=t) और (12x+16y=20) में कोई हल न हो, इसके लिए (t) पर कौन-सी शर्त होगी?
For the pair (3x+4y=t) and (12x+16y=20) to have no solution, what condition is required on (t)?
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A (t=5)
B (t=20)
C (t=4)
D \(t\neq5\)
Explanation opens after your attempt
Correct Answer
D. \(t\neq5\)
Step 1
Concept
The coefficient ratio is \(\frac{1}{4}\). For no solution, \(\frac{t}{20}\neq\frac{1}{4}\), so \(t\neq5\).
Step 2
Why this answer is correct
The correct answer is D. \(t\neq5\). The coefficient ratio is \(\frac{1}{4}\). For no solution, \(\frac{t}{20}\neq\frac{1}{4}\), so \(t\neq5\).
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{4}\) है। कोई हल नहीं के लिए \(\frac{t}{20}\neq\frac{1}{4}\), इसलिए \(t\neq5\)।
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युग्म (11x+2y=3) और (22x+ny=8) का अद्वितीय हल कब होगा?
When will (11x+2y=3) and (22x+ny=8) have a unique solution?
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#solvability
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A \(n\neq4\)
B (n=4)
C (n=2)
D (n=8)
Explanation opens after your attempt
Correct Answer
A. \(n\neq4\)
Step 1
Concept
For a unique solution, \(\frac{11}{22}\neq\frac{2}{n}\) is required. Hence \(n\neq4\) is the correct exam condition.
Step 2
Why this answer is correct
The correct answer is A. \(n\neq4\). For a unique solution, \(\frac{11}{22}\neq\frac{2}{n}\) is required. Hence \(n\neq4\) is the correct exam condition.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{11}{22}\neq\frac{2}{n}\) चाहिए। अतः \(n\neq4\) परीक्षा में सही शर्त है।
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रेखाएँ (2x-5y=1) और (6x-15y=3) किस प्रकार का हल देती हैं?
What type of solution is given by the lines (2x-5y=1) and (6x-15y=3)?
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A कोई हल नहीं / No solution
B अद्वितीय हल / Unique solution
C अनंत हल / Infinitely many solutions
D केवल दो हल / Exactly two solutions
Explanation opens after your attempt
Correct Answer
C. अनंत हल / Infinitely many solutions
Step 1
Concept
All three ratios are equal, so the lines are coincident. Coincident lines give infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. All three ratios are equal, so the lines are coincident. Coincident lines give infinitely many solutions.
Step 3
Exam Tip
तीनों अनुपात समान हैं, इसलिए रेखाएँ संपाती हैं। संपाती रेखाएँ अनंत हल देती हैं।
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रेखाएँ (7x+3y=10) और (14x+6y=19) के लिए सही निष्कर्ष क्या है?
What is the correct conclusion for the lines (7x+3y=10) and (14x+6y=19)?
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A अनंत हल / Infinitely many solutions
B कोई हल नहीं / No solution
C अद्वितीय हल / Unique solution
D निर्धारित नहीं / Cannot be decided
Explanation opens after your attempt
Correct Answer
B. कोई हल नहीं / No solution
Step 1
Concept
Coefficient ratios are equal but the constant ratio is different. Hence the lines are parallel and have no solution.
Step 2
Why this answer is correct
The correct answer is B. कोई हल नहीं / No solution. Coefficient ratios are equal but the constant ratio is different. Hence the lines are parallel and have no solution.
Step 3
Exam Tip
गुणांकों के अनुपात समान हैं पर स्थिर पद का अनुपात अलग है। इसलिए रेखाएँ समांतर हैं और कोई हल नहीं है।
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युग्म (4x-7y=5) और (12x-21y=17) की हल-स्थिति क्या है?
What is the solution status of (4x-7y=5) and (12x-21y=17)?
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A अद्वितीय हल / Unique solution
B अनंत हल / Infinitely many solutions
C कोई हल नहीं / No solution
D तीन हल / Three solutions
Explanation opens after your attempt
Correct Answer
C. कोई हल नहीं / No solution
Step 1
Concept
The coefficient ratio is \(\frac{1}{3}\) but the constant ratio is \(\frac{5}{17}\). Therefore, there is no solution.
Step 2
Why this answer is correct
The correct answer is C. कोई हल नहीं / No solution. The coefficient ratio is \(\frac{1}{3}\) but the constant ratio is \(\frac{5}{17}\). Therefore, there is no solution.
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{3}\) है पर स्थिर अनुपात \(\frac{5}{17}\) है। अतः कोई हल नहीं मिलेगा।
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युग्म (2x+3y=5) और (4x+6y=10) के लिए कौन-सा कथन सही है?
Which statement is correct for (2x+3y=5) and (4x+6y=10)?
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A कोई हल नहीं / No solution
B अद्वितीय हल / Unique solution
C हल केवल (x=0) पर है / Solution only at (x=0)
D अनंत हल / Infinitely many solutions
Explanation opens after your attempt
Correct Answer
D. अनंत हल / Infinitely many solutions
Step 1
Concept
The second equation is (2) times the first. Thus both are the same line and have infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is D. अनंत हल / Infinitely many solutions. The second equation is (2) times the first. Thus both are the same line and have infinitely many solutions.
Step 3
Exam Tip
दूसरा समीकरण पहले का (2) गुना है। इसलिए दोनों एक ही रेखा हैं और अनंत हल हैं।
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युग्म (9x-4y=11) और (18x-8y=21) की प्रकृति क्या है?
What is the nature of the pair (9x-4y=11) and (18x-8y=21)?
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A कोई हल नहीं / No solution
B अनंत हल / Infinitely many solutions
C अद्वितीय हल / Unique solution
D हर मान हल है / Every value is a solution
Explanation opens after your attempt
Correct Answer
A. कोई हल नहीं / No solution
Step 1
Concept
The coefficient ratio is equal but the constant ratio is not equal. In exams, treat this as distinct parallel lines.
Step 2
Why this answer is correct
The correct answer is A. कोई हल नहीं / No solution. The coefficient ratio is equal but the constant ratio is not equal. In exams, treat this as distinct parallel lines.
Step 3
Exam Tip
गुणांक अनुपात समान है लेकिन स्थिर पद का अनुपात समान नहीं है। परीक्षा में इसे समांतर अलग रेखाओं का संकेत मानें।
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युग्म (5x+2y=6) और (15x+7y=18) के लिए हलों की संख्या क्या है?
How many solutions does the pair (5x+2y=6) and (15x+7y=18) have?
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A कोई हल नहीं / No solution
B अनंत हल / Infinitely many solutions
C अद्वितीय हल / Unique solution
D ठीक (2) हल / Exactly (2) solutions
Explanation opens after your attempt
Correct Answer
C. अद्वितीय हल / Unique solution
Step 1
Concept
Here \(5\times7-15\times2\neq0\). Therefore, the pair has a unique solution.
Step 2
Why this answer is correct
The correct answer is C. अद्वितीय हल / Unique solution. Here \(5\times7-15\times2\neq0\). Therefore, the pair has a unique solution.
Step 3
Exam Tip
यहाँ \(5\times7-15\times2\neq0\) है। इसलिए युग्म का अद्वितीय हल है।
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युग्म ((k+1)x+4y=9) और (6x+8y=15) में कोई हल न होने के लिए (k) क्या होगा?
For no solution in ((k+1)x+4y=9) and (6x+8y=15), what is (k)?
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A (k=1)
B (k=2)
C (k=3)
D (k=5)
Explanation opens after your attempt
Step 1
Concept
Making coefficient ratios equal gives (k=2). The constant ratio is different, so there is no solution.
Step 2
Why this answer is correct
The correct answer is B. (k=2). Making coefficient ratios equal gives (k=2). The constant ratio is different, so there is no solution.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर (k=2) आता है। स्थिर अनुपात अलग है, इसलिए कोई हल नहीं होगा।
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युग्म (7x+(m-4 )y=13) और (14x+6y=26) के अनंत हलों के लिए (m) का मान बताइए।
Find (m) for infinitely many solutions of (7x+(m-4 )y=13) and (14x+6y=26).
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A (m=6)
B (m=7)
C (m=8)
D (m=10)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{7}{14}=\frac{m-4}{6}=\frac{13}{26}\) must hold. This gives (m=7).
Step 2
Why this answer is correct
The correct answer is B. (m=7). For infinitely many solutions, \(\frac{7}{14}=\frac{m-4}{6}=\frac{13}{26}\) must hold. This gives (m=7).
Step 3
Exam Tip
अनंत हलों में \(\frac{7}{14}=\frac{m-4}{6}=\frac{13}{26}\) होना चाहिए। इससे (m=7) मिलता है।
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युग्म ((p-2)x+9y=4) और (5x+15y=12) का अद्वितीय हल कब होगा?
When will ((p-2)x+9y=4) and (5x+15y=12) have a unique solution?
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#linear-equations
#solvability
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A (p=5)
B (p=2)
C \(p\neq5\)
D \(p\neq2\)
Explanation opens after your attempt
Correct Answer
C. \(p\neq5\)
Step 1
Concept
For a unique solution, \(\frac{p-2}{5}\neq\frac{9}{15}\) is needed. Hence \(p\neq5\) is correct.
Step 2
Why this answer is correct
The correct answer is C. \(p\neq5\). For a unique solution, \(\frac{p-2}{5}\neq\frac{9}{15}\) is needed. Hence \(p\neq5\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{p-2}{5}\neq\frac{9}{15}\) चाहिए। इसलिए \(p\neq5\) सही है।
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युग्म (3x+ay=6) और (9x+12y=b) के अनंत हलों के लिए ((a,b)) क्या होगा?
For infinitely many solutions of (3x+ay=6) and (9x+12y=b), what is ((a,b))?
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#linear-equations
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A (a=3,\ b=12)
B (a=4,\ b=18)
C (a=6,\ b=18)
D (a=4,\ b=12)
Explanation opens after your attempt
Correct Answer
B. (a=4,\ b=18)
Step 1
Concept
All three ratios must be \(\frac{1}{3}\). So (a=4) and (b=18).
Step 2
Why this answer is correct
The correct answer is B. (a=4,\ b=18). All three ratios must be \(\frac{1}{3}\). So (a=4) and (b=18).
Step 3
Exam Tip
तीनों अनुपात \(\frac{1}{3}\) होने चाहिए। इसलिए (a=4) और (b=18) है।
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युग्म (5x-2y=c) और (15x-6y=21) में कोई हल न हो, इसके लिए (c) की शर्त क्या है?
For no solution in (5x-2y=c) and (15x-6y=21), what condition should (c) satisfy?
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A (c=7)
B (c=21)
C \(c\neq7\)
D \(c\neq21\)
Explanation opens after your attempt
Correct Answer
C. \(c\neq7\)
Step 1
Concept
The coefficient ratio is \(\frac{1}{3}\). For no solution, \(\frac{c}{21}\neq\frac{1}{3}\), hence \(c\neq7\).
Step 2
Why this answer is correct
The correct answer is C. \(c\neq7\). The coefficient ratio is \(\frac{1}{3}\). For no solution, \(\frac{c}{21}\neq\frac{1}{3}\), hence \(c\neq7\).
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{3}\) है। कोई हल नहीं के लिए \(\frac{c}{21}\neq\frac{1}{3}\), अतः \(c\neq7\)।
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युग्म (rx+5y=2) और (8x+10y=4) का अद्वितीय हल कब होगा?
When will (rx+5y=2) and (8x+10y=4) have a unique solution?
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A (r=4)
B \(r\neq4\)
C (r=8)
D \(r\neq8\)
Explanation opens after your attempt
Correct Answer
B. \(r\neq4\)
Step 1
Concept
For a unique solution, \(\frac{r}{8}\neq\frac{5}{10}\) must hold. So \(r\neq4\) is correct.
Step 2
Why this answer is correct
The correct answer is B. \(r\neq4\). For a unique solution, \(\frac{r}{8}\neq\frac{5}{10}\) must hold. So \(r\neq4\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{r}{8}\neq\frac{5}{10}\) होना चाहिए। इसलिए \(r\neq4\) सही है।
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युग्म (12x+sy=18) और (20x+10y=30) के अनंत हलों के लिए (s) क्या होगा?
What is (s) for infinitely many solutions of (12x+sy=18) and (20x+10y=30)?
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A (s=4)
B (s=5)
C (s=6)
D (s=8)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{12}{20}=\frac{s}{10}=\frac{18}{30}\) is needed. This gives (s=6).
Step 2
Why this answer is correct
The correct answer is C. (s=6). For infinitely many solutions, \(\frac{12}{20}=\frac{s}{10}=\frac{18}{30}\) is needed. This gives (s=6).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{12}{20}=\frac{s}{10}=\frac{18}{30}\) चाहिए। इससे (s=6) आता है।
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युग्म (ux+9y=7) और (4x+6y=11) में कोई हल न होने के लिए (u) का मान क्या है?
For no solution in (ux+9y=7) and (4x+6y=11), what is the value of (u)?
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A (u=4)
B (u=5)
C (u=6)
D (u=9)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios, \(\frac{u}{4}=\frac{9}{6}\) gives (u=6). The constant ratio is different, so there is no solution.
Step 2
Why this answer is correct
The correct answer is C. (u=6). Equating coefficient ratios, \(\frac{u}{4}=\frac{9}{6}\) gives (u=6). The constant ratio is different, so there is no solution.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{u}{4}=\frac{9}{6}\) से (u=6) मिलता है। स्थिर पद अनुपात अलग है, इसलिए कोई हल नहीं है।
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युग्म (13x-ty=5) और (26x-10y=14) का अद्वितीय हल कब होगा?
When will (13x-ty=5) and (26x-10y=14) have a unique solution?
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A (t=5)
B (t=10)
C \(t\neq10\)
D \(t\neq5\)
Explanation opens after your attempt
Correct Answer
D. \(t\neq5\)
Step 1
Concept
For a unique solution, \(\frac{13}{26}\neq\frac{t}{10}\) must hold. Hence \(t\neq5\) is correct.
Step 2
Why this answer is correct
The correct answer is D. \(t\neq5\). For a unique solution, \(\frac{13}{26}\neq\frac{t}{10}\) must hold. Hence \(t\neq5\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{13}{26}\neq\frac{t}{10}\) होना चाहिए। अतः \(t\neq5\) सही है।
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युग्म ((v+2)x-3y=6) और (10x-5y=10) के अनंत हलों के लिए (v) क्या होगा?
For infinitely many solutions of ((v+2)x-3y=6) and (10x-5y=10), what is (v)?
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A (v=2)
B (v=3)
C (v=4)
D (v=5)
Explanation opens after your attempt
Step 1
Concept
Here \(\frac{v+2}{10}=\frac{-3}{-5}=\frac{6}{10}\) must hold. This gives (v=4).
Step 2
Why this answer is correct
The correct answer is C. (v=4). Here \(\frac{v+2}{10}=\frac{-3}{-5}=\frac{6}{10}\) must hold. This gives (v=4).
Step 3
Exam Tip
यहाँ \(\frac{v+2}{10}=\frac{-3}{-5}=\frac{6}{10}\) होना चाहिए। इससे (v=4) मिलता है।
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युग्म (2x+3y=d) और (14x+21y=28) में कोई हल न हो, इसके लिए (d) पर कौन-सी शर्त होगी?
For (2x+3y=d) and (14x+21y=28) to have no solution, what condition is required on (d)?
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A (d=4)
B \(d\neq4\)
C (d=7)
D \(d\neq7\)
Explanation opens after your attempt
Correct Answer
B. \(d\neq4\)
Step 1
Concept
The coefficient ratio is \(\frac{1}{7}\). For no solution, \(\frac{d}{28}\neq\frac{1}{7}\), so \(d\neq4\).
Step 2
Why this answer is correct
The correct answer is B. \(d\neq4\). The coefficient ratio is \(\frac{1}{7}\). For no solution, \(\frac{d}{28}\neq\frac{1}{7}\), so \(d\neq4\).
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{7}\) है। कोई हल नहीं के लिए \(\frac{d}{28}\neq\frac{1}{7}\), इसलिए \(d\neq4\)।
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युग्म (ax+by=1) और (2ax+3by=5) के अद्वितीय हल की शर्त क्या है?
What is the condition for a unique solution of (ax+by=1) and (2ax+3by=5)?
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A (ab=0)
B (a+b=0)
C \(ab\neq0\)
D (a=b)
Explanation opens after your attempt
Correct Answer
C. \(ab\neq0\)
Step 1
Concept
The determinant is (D=3ab-2ab=ab). For a unique solution, \(ab\neq0\) is needed.
Step 2
Why this answer is correct
The correct answer is C. \(ab\neq0\). The determinant is (D=3ab-2ab=ab). For a unique solution, \(ab\neq0\) is needed.
Step 3
Exam Tip
सारणिक (D=3ab-2ab=ab) है। अद्वितीय हल के लिए \(ab\neq0\) चाहिए।
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युग्म (lx+6y=8) और (9x+18y=10) में कोई हल न होने के लिए (l) क्या होगा?
What is (l) for no solution in (lx+6y=8) and (9x+18y=10)?
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A (l=2)
B (l=3)
C (l=6)
D (l=9)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios, \(\frac{l}{9}=\frac{6}{18}\) gives (l=3). The constant ratio is different.
Step 2
Why this answer is correct
The correct answer is B. (l=3). Equating coefficient ratios, \(\frac{l}{9}=\frac{6}{18}\) gives (l=3). The constant ratio is different.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{l}{9}=\frac{6}{18}\) से (l=3) मिलता है। स्थिर पद अनुपात अलग है।
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युग्म (6x+(h+1)y=12) और (15x+20y=30) के अनंत हलों के लिए (h) क्या है?
For infinitely many solutions of (6x+(h+1)y=12) and (15x+20y=30), what is (h)?
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A (h=5)
B (h=6)
C (h=7)
D (h=8)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{6}{15}=\frac{h+1}{20}=\frac{12}{30}\) must hold. This gives (h=7).
Step 2
Why this answer is correct
The correct answer is C. (h=7). For infinitely many solutions, \(\frac{6}{15}=\frac{h+1}{20}=\frac{12}{30}\) must hold. This gives (h=7).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{6}{15}=\frac{h+1}{20}=\frac{12}{30}\) होना चाहिए। इससे (h=7) मिलता है।
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युग्म (x+2y=3) और (4x+8y=12) के लिए सही हल-स्थिति कौन-सी है?
Which solution status is correct for (x+2y=3) and (4x+8y=12)?
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A अनंत हल / Infinitely many solutions
B कोई हल नहीं / No solution
C अद्वितीय हल / Unique solution
D एक भी बिंदु नहीं / No point at all
Explanation opens after your attempt
Correct Answer
A. अनंत हल / Infinitely many solutions
Step 1
Concept
The second equation is (4) times the first. Both are the same line, so there are infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is A. अनंत हल / Infinitely many solutions. The second equation is (4) times the first. Both are the same line, so there are infinitely many solutions.
Step 3
Exam Tip
दूसरा समीकरण पहले का (4) गुना है। दोनों समान रेखाएँ हैं, इसलिए अनंत हल हैं।
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युग्म (3x-y=4) और (6x-2y=9) के बारे में सही कथन चुनिए।
Choose the correct statement about (3x-y=4) and (6x-2y=9).
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A अद्वितीय हल है / It has a unique solution
B अनंत हल हैं / It has infinitely many solutions
C कोई हल नहीं है / It has no solution
D दो हल हैं / It has two solutions
Explanation opens after your attempt
Correct Answer
C. कोई हल नहीं है / It has no solution
Step 1
Concept
The coefficient ratio is equal but the constant ratio is not equal. Hence the lines are parallel and distinct.
Step 2
Why this answer is correct
The correct answer is C. कोई हल नहीं है / It has no solution. The coefficient ratio is equal but the constant ratio is not equal. Hence the lines are parallel and distinct.
Step 3
Exam Tip
गुणांक अनुपात समान है लेकिन स्थिर पद अनुपात समान नहीं है। इसलिए रेखाएँ समांतर और अलग हैं।
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युग्म (4x+5y=1) और (8x+9y=2) के लिए हलों की संख्या क्या है?
How many solutions does (4x+5y=1) and (8x+9y=2) have?
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A कोई हल नहीं / No solution
B अनंत हल / Infinitely many solutions
C अद्वितीय हल / Unique solution
D निर्भर हल / Dependent solutions
Explanation opens after your attempt
Correct Answer
C. अद्वितीय हल / Unique solution
Step 1
Concept
Here \(\frac{4}{8}\neq\frac{5}{9}\). Therefore, the lines intersect at one point.
Step 2
Why this answer is correct
The correct answer is C. अद्वितीय हल / Unique solution. Here \(\frac{4}{8}\neq\frac{5}{9}\). Therefore, the lines intersect at one point.
Step 3
Exam Tip
यहाँ \(\frac{4}{8}\neq\frac{5}{9}\) है। इसलिए रेखाएँ एक बिंदु पर कटती हैं।
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सामान्य युग्म \(a_1x+b_1y+c_1=0\) और \(a_2x+b_2y+c_2=0\) में कोई हल न होने की शर्त क्या है?
For the general pair \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), what is the condition for no solution?
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A \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\)
B \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)
C \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\)
D \(a_1=a_2=0\)
Explanation opens after your attempt
Correct Answer
C. \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\)
Step 1
Concept
No solution occurs when lines are parallel and distinct. Its ratio form is \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\).
Step 2
Why this answer is correct
The correct answer is C. \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\). No solution occurs when lines are parallel and distinct. Its ratio form is \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\).
Step 3
Exam Tip
कोई हल नहीं तब होता है जब रेखाएँ समांतर और अलग हों। इसका अनुपात रूप \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\) है।
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यदि किसी युग्म में गुणांक अनुपात समान है और स्थिर पद का अनुपात अलग है, तो हल-स्थिति क्या होगी?
If coefficient ratios are equal and the constant ratio is different in a pair, what is the solution status?
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A अद्वितीय हल / Unique solution
B अनंत हल / Infinitely many solutions
C कोई हल नहीं / No solution
D सदैव दो हल / Always two solutions
Explanation opens after your attempt
Correct Answer
C. कोई हल नहीं / No solution
Step 1
Concept
This condition forms distinct parallel lines. Therefore, the pair is inconsistent.
Step 2
Why this answer is correct
The correct answer is C. कोई हल नहीं / No solution. This condition forms distinct parallel lines. Therefore, the pair is inconsistent.
Step 3
Exam Tip
यह स्थिति अलग-अलग समांतर रेखाएँ बनाती है। इसलिए युग्म असंगत होता है।
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यदि \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\) है, तो दो रैखिक समीकरणों के युग्म के लिए क्या निष्कर्ष होगा?
If \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\), what is the conclusion for a pair of two linear equations?
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A कोई हल नहीं / No solution
B अनंत हल / Infinitely many solutions
C अद्वितीय हल / Unique solution
D हल निर्भर करता है / Solution depends only on constants
Explanation opens after your attempt
Correct Answer
C. अद्वितीय हल / Unique solution
Step 1
Concept
When coefficient ratios are different, the lines intersect at one point. Therefore, a unique solution is obtained.
Step 2
Why this answer is correct
The correct answer is C. अद्वितीय हल / Unique solution. When coefficient ratios are different, the lines intersect at one point. Therefore, a unique solution is obtained.
Step 3
Exam Tip
गुणांक अनुपात अलग होने पर रेखाएँ एक बिंदु पर कटती हैं। इसलिए अद्वितीय हल मिलता है।
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संपाती रेखाओं के लिए सही बीजीय शर्त कौन-सी है?
Which algebraic condition is correct for coincident lines?
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A \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\)
B \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\)
C \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)
D \(a_1b_2-a_2b_1\neq0\)
Explanation opens after your attempt
Correct Answer
C. \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)
Step 1
Concept
Coincident lines represent the same line. Hence all three ratios are equal and infinitely many solutions occur.
Step 2
Why this answer is correct
The correct answer is C. \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\). Coincident lines represent the same line. Hence all three ratios are equal and infinitely many solutions occur.
Step 3
Exam Tip
संपाती रेखाएँ एक ही रेखा को दर्शाती हैं। इसलिए तीनों अनुपात समान होते हैं और अनंत हल मिलते हैं।
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यदि (D=0) और कम से कम एक सहायक सारणिक शून्य नहीं है, तो युग्म की हल-स्थिति क्या होगी?
If (D=0) and at least one auxiliary determinant is non-zero, what is the solution status of the pair?
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A अनंत हल / Infinitely many solutions
B अद्वितीय हल / Unique solution
C कोई हल नहीं / No solution
D सदैव हल / Always solvable
Explanation opens after your attempt
Correct Answer
C. कोई हल नहीं / No solution
Step 1
Concept
(D=0) with a non-zero auxiliary determinant indicates distinct parallel lines. Hence there is no solution.
Step 2
Why this answer is correct
The correct answer is C. कोई हल नहीं / No solution. (D=0) with a non-zero auxiliary determinant indicates distinct parallel lines. Hence there is no solution.
Step 3
Exam Tip
(D=0) और असंगत सहायक सारणिक समांतर अलग रेखाओं का संकेत देते हैं। इसलिए कोई हल नहीं होता।
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यदि दो रेखाओं की ढाल समान और (y)-अवरोध अलग हों, तो उनके समीकरणों के युग्म में कितने हल होंगे?
If two lines have the same slope and different (y)-intercepts, how many solutions will their pair of equations have?
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A अद्वितीय हल / Unique solution
B अनंत हल / Infinitely many solutions
C कोई हल नहीं / No solution
D दो हल / Two solutions
Explanation opens after your attempt
Correct Answer
C. कोई हल नहीं / No solution
Step 1
Concept
Same slope and different intercepts mean distinct parallel lines. Therefore, they never intersect.
Step 2
Why this answer is correct
The correct answer is C. कोई हल नहीं / No solution. Same slope and different intercepts mean distinct parallel lines. Therefore, they never intersect.
Step 3
Exam Tip
समान ढाल और अलग अवरोध का अर्थ अलग समांतर रेखाएँ है। इसलिए वे कभी नहीं कटतीं।
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यदि दो रेखाओं की ढाल अलग-अलग हों, तो उनके युग्म के लिए कौन-सा निष्कर्ष सही है?
If two lines have different slopes, which conclusion is correct for their pair?
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#linear-equations
#solvability
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A कोई हल नहीं / No solution
B अनंत हल / Infinitely many solutions
C अद्वितीय हल / Unique solution
D समीकरण निर्भर हैं / Equations are dependent
Explanation opens after your attempt
Correct Answer
C. अद्वितीय हल / Unique solution
Step 1
Concept
Lines with different slopes meet at exactly one point. Hence the pair is consistent and independent.
Step 2
Why this answer is correct
The correct answer is C. अद्वितीय हल / Unique solution. Lines with different slopes meet at exactly one point. Hence the pair is consistent and independent.
Step 3
Exam Tip
अलग ढाल वाली रेखाएँ केवल एक बिंदु पर मिलती हैं। इसलिए युग्म संगत और स्वतंत्र होता है।
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युग्म ((k-3)x+2y=5) और (4x+ky=11) अद्वितीय न हो, इसके लिए (k) क्या हो सकता है?
For ((k-3)x+2y=5) and (4x+ky=11) to be non-unique, what can (k) be?
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A \(k=\frac{3\pm\sqrt{41}}{2}\)
B \(k=\frac{-3\pm\sqrt{41}}{2}\)
C (k=3)
D (k=4)
Explanation opens after your attempt
Correct Answer
A. \(k=\frac{3\pm\sqrt{41}}{2}\)
Step 1
Concept
For non-unique solutions, the determinant must be zero. From ((k-3)k-8=0), \(k=\frac{3\pm\sqrt{41}}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(k=\frac{3\pm\sqrt{41}}{2}\). For non-unique solutions, the determinant must be zero. From ((k-3)k-8=0), \(k=\frac{3\pm\sqrt{41}}{2}\).
Step 3
Exam Tip
अद्वितीय न होने के लिए सारणिक शून्य होना चाहिए। ((k-3)k-8=0) से \(k=\frac{3\pm\sqrt{41}}{2}\) मिलता है।
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युग्म ((k+1)x+6y=3) और (2x+12y=8) के लिए कोई हल न होने पर (k) का मान क्या होगा?
For no solution in ((k+1)x+6y=3) and (2x+12y=8), what is (k)?
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A (k=-1)
B (k=0)
C (k=1)
D (k=2)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios gives (k=0). The constant ratio \(\frac{3}{8}\) is different, so there is no solution.
Step 2
Why this answer is correct
The correct answer is B. (k=0). Equating coefficient ratios gives (k=0). The constant ratio \(\frac{3}{8}\) is different, so there is no solution.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर (k=0) मिलता है। स्थिर अनुपात \(\frac{3}{8}\) अलग है, इसलिए कोई हल नहीं।
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युग्म (4x+(m-5 )y=9) और (12x+6y=27) के अनंत हलों के लिए (m) कितना है?
What is (m) for infinitely many solutions of (4x+(m-5 )y=9) and (12x+6y=27)?
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A (m=5)
B (m=6)
C (m=7)
D (m=9)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{4}{12}=\frac{m-5}{6}=\frac{9}{27}\) is required. This gives (m=7).
Step 2
Why this answer is correct
The correct answer is C. (m=7). For infinitely many solutions, \(\frac{4}{12}=\frac{m-5}{6}=\frac{9}{27}\) is required. This gives (m=7).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{4}{12}=\frac{m-5}{6}=\frac{9}{27}\) चाहिए। इससे (m=7) मिलता है।
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युग्म ((p+4)x-9y=2) और (5x-15y=6) का अद्वितीय हल कब होगा?
When will ((p+4)x-9y=2) and (5x-15y=6) have a unique solution?
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A (p=-1)
B (p=1)
C \(p\neq-1\)
D \(p\neq1\)
Explanation opens after your attempt
Correct Answer
C. \(p\neq-1\)
Step 1
Concept
For a unique solution, \(\frac{p+4}{5}\neq\frac{-9}{-15}\) must hold. Hence \(p\neq-1\).
Step 2
Why this answer is correct
The correct answer is C. \(p\neq-1\). For a unique solution, \(\frac{p+4}{5}\neq\frac{-9}{-15}\) must hold. Hence \(p\neq-1\).
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{p+4}{5}\neq\frac{-9}{-15}\) होना चाहिए। इसलिए \(p\neq-1\)।
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युग्म \(\frac{1}{2}x+qy=3\) और (2x+8y=5) के लिए कोई हल न होने पर (q) क्या होगा?
For no solution in \(\frac{1}{2}x+qy=3\) and (2x+8y=5), what is (q)?
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A (q=1)
B (q=2)
C (q=4)
D (q=8)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios gives (q=2). The constant ratio is not equal, so there is no solution.
Step 2
Why this answer is correct
The correct answer is B. (q=2). Equating coefficient ratios gives (q=2). The constant ratio is not equal, so there is no solution.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर (q=2) मिलता है। स्थिर पद अनुपात समान नहीं है, इसलिए कोई हल नहीं।
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युग्म \(rx+\frac{3}{2}y=6\) और (8x+6y=24) के अनंत हलों के लिए (r) का मान क्या है?
What is (r) for infinitely many solutions of \(rx+\frac{3}{2}y=6\) and (8x+6y=24)?
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A (r=1)
B (r=2)
C (r=3)
D (r=4)
Explanation opens after your attempt
Step 1
Concept
All three ratios must be \(\frac{1}{4}\). Therefore, (r=2) is the correct value.
Step 2
Why this answer is correct
The correct answer is B. (r=2). All three ratios must be \(\frac{1}{4}\). Therefore, (r=2) is the correct value.
Step 3
Exam Tip
तीनों अनुपात \(\frac{1}{4}\) होने चाहिए। इसलिए (r=2) सही मान है।
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युग्म \(\alpha x+4y=7\) और (9x+12y=21) का अद्वितीय हल कब होगा?
When will \(\alpha x+4y=7\) and (9x+12y=21) have a unique solution?
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A \(\alpha=3\)
B \(\alpha\neq3\)
C \(\alpha=9\)
D \(\alpha\neq9\)
Explanation opens after your attempt
Correct Answer
B. \(\alpha\neq3\)
Step 1
Concept
For a unique solution, \(\frac{\alpha}{9}\neq\frac{4}{12}\) must hold. Thus \(\alpha\neq3\) is correct.
Step 2
Why this answer is correct
The correct answer is B. \(\alpha\neq3\). For a unique solution, \(\frac{\alpha}{9}\neq\frac{4}{12}\) must hold. Thus \(\alpha\neq3\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{\alpha}{9}\neq\frac{4}{12}\) होना चाहिए। इसलिए \(\alpha\neq3\) सही शर्त है।
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युग्म (3x+\(\beta-1\)y=10) और (6x+4y=18) में कोई हल न होने के लिए \(\beta\) क्या होगा?
For no solution in (3x+\(\beta-1\)y=10) and (6x+4y=18), what is \(\beta\)?
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A \(\beta=1\)
B \(\beta=2\)
C \(\beta=3\)
D \(\beta=4\)
Explanation opens after your attempt
Correct Answer
C. \(\beta=3\)
Step 1
Concept
Equating coefficient ratios gives \(\beta=3\). The constant ratio \(\frac{5}{9}\) is different, so there is no solution.
Step 2
Why this answer is correct
The correct answer is C. \(\beta=3\). Equating coefficient ratios gives \(\beta=3\). The constant ratio \(\frac{5}{9}\) is different, so there is no solution.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\beta=3\) मिलता है। स्थिर अनुपात \(\frac{5}{9}\) अलग है, इसलिए कोई हल नहीं।
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युग्म \(5x-2y=\gamma\) और (20x-8y=36) के अनंत हलों के लिए \(\gamma\) का मान क्या है?
What is \(\gamma\) for infinitely many solutions of \(5x-2y=\gamma\) and (20x-8y=36)?
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A \(\gamma=6\)
B \(\gamma=8\)
C \(\gamma=9\)
D \(\gamma=12\)
Explanation opens after your attempt
Correct Answer
C. \(\gamma=9\)
Step 1
Concept
The coefficient ratio is \(\frac{1}{4}\). For infinitely many solutions, \(\frac{\gamma}{36}=\frac{1}{4}\), so \(\gamma=9\).
Step 2
Why this answer is correct
The correct answer is C. \(\gamma=9\). The coefficient ratio is \(\frac{1}{4}\). For infinitely many solutions, \(\frac{\gamma}{36}=\frac{1}{4}\), so \(\gamma=9\).
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{4}\) है। अनंत हलों के लिए \(\frac{\gamma}{36}=\frac{1}{4}\), इसलिए \(\gamma=9\)।
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युग्म ((s+2)x+3y=1) और (5x+(s-1 )y=4) के अद्वितीय हल की सही शर्त कौन-सी है?
Which condition gives a unique solution for ((s+2)x+3y=1) and (5x+(s-1 )y=4)?
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A \(s^2+s-17=0\)
B \(s^2+s-17\neq0\)
C \(s^2-s-17=0\)
D \(s^2-s-17\neq0\)
Explanation opens after your attempt
Correct Answer
B. \(s^2+s-17\neq0\)
Step 1
Concept
The determinant is (D=(s+2)(s-1 )-15=s-2 +s-17 ). For a unique solution, \(D\neq0\) is required.
Step 2
Why this answer is correct
The correct answer is B. \(s^2+s-17\neq0\). The determinant is (D=(s+2)(s-1 )-15=s-2 +s-17 ). For a unique solution, \(D\neq0\) is required.
Step 3
Exam Tip
सारणिक (D=(s+2)(s-1 )-15=s-2 +s-17 ) है। अद्वितीय हल के लिए \(D\neq0\) चाहिए।
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